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i A A p p r r o o b b l l e e m m - - p p o o s s i i n n g g i i n n t t e e r r v v e e n n t t i i o o n n i i n n t t h h e e d d e e v v e e l l o o p p m m e e n n t t o o f f p p r r o o b b l l e e m m - - s s o o l l v v i i n n g g c c o o m m p p e e t t e e n n c c e e o o f f u u n n d d e e r r a a c c h h i i e e v v i i n n g g , , m m i i d d d d l l e e - - y y e e a a r r s s t t u u d d e e n n t t s s . . Deborah Jean Priest M. Ed. (QUT), BSc Ed. (Melb.) Faculty of Education Queensland University of Technology Kelvin Grove Campus, Brisbane. A Thesis submitted in fulfilment of the requirements leading to the award of the degree of Doctor of Philosophy May 2009

A problem-posing intervention - QUT · posing skills development and improvements in problem-solving competence. A cohort of Year 7 students at a private, non-denominational, co-educational

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  • i

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    Deborah Jean Priest

    M. Ed. (QUT), BSc Ed. (Melb.)

    Faculty of Education

    Queensland University of Technology

    Kelvin Grove Campus, Brisbane.

    A Thesis submitted in fulfilment of the requirements leading to the award of

    the degree of Doctor of Philosophy

    May 2009

  • ii

    CERTIFICATE RECOMMENDING ACCEPTANCE

  • iii

    DEFINITION OF ACRONYMS

    ACER Australian Council for Educational Research

    ANTA Australian National Training Authority

    DEST Department of Education, Science and Training

    GSA Graduate Skills Assessment Test

    IQ Intelligence Quotient

    MCEETYA Ministerial Council on Education, Employment, Training and

    Youth Affairs

    MYAT Middle Years Ability Test

    NAPLAN National Assessment Program: Literacy and Numeracy

    NCB National Curriculum Board

    NCTM National Council of Teachers of Mathematics

    (United States of America)

    NNR National Numeracy Review

    NMAP National Mathematics Advisory Panel

    (United States of America)

    NRC National Research Council (United States of America)

    POPS Profiles of Problem Solving Test

  • iv

    KEYW ORDS

    assessment

    cognition

    developmental learning

    education

    engagement

    intervention

    mathematics

    middle year

    multiple intelligences

    pedagogy

    problem solving

    problem posing

    self-regulation

    teaching experiment

    underachievement

  • iv

    ABSTRACT

    This study reported on the issues surrounding the acquisition of problem-

    solving competence of middle-year students who had been ascertained as

    above average in intelligence, but underachieving in problem-solving

    competence. In particular, it looked at the possible links between problem-

    posing skills development and improvements in problem-solving

    competence.

    A cohort of Year 7 students at a private, non-denominational, co-educational

    school was chosen as participants for the study, as they undertook a series

    of problem-posing sessions each week throughout a school term. The

    lessons were facilitated by the researcher in the students’ school setting.

    Two criteria were chosen to identify participants for this study. Firstly, each

    participant scored above the 60th percentile in the standardized Middle Years

    Ability Test (MYAT) (Australian Council for Educational Research, 2005) and

    secondly, the participants all scored below the cohort average for Criterion B

    (Problem-solving Criterion) in their school mathematics tests during the first

    semester of Year 7.

    Two mutually exclusive groups of participants were investigated with one

    constituting the Comparison Group and the other constituting the Intervention

    Group. The Comparison Group was chosen from a Year 7 cohort for whom

    no problem-posing intervention had occurred, while the Intervention Group

    was chosen from the Year 7 cohort of the following year. This second group

    received the problem-posing intervention in the form of a teaching

    experiment. That is, the Comparison Group were only pre-tested and post-

    tested, while the Intervention Group was involved in the teaching experiment

    and received the pre-testing and post-testing at the same time of the year,

    but in the following year, when the Comparison Group have moved on to the

    secondary part of the school. The groups were chosen from consecutive

    Year 7 cohorts to avoid cross-contamination of the data.

  • v

    A constructionist framework was adopted for this study that allowed the

    researcher to gain an “authentic understanding” of the changes that occurred

    in the development of problem-solving competence of the participants in the

    context of a classroom setting (Richardson, 1999). Qualitative and

    quantitative data were collected through a combination of methods including

    researcher observation and journal writing, video taping, student workbooks,

    informal student interviews, student surveys, and pre-testing and post-

    testing. This combination of methods was required to increase the validity of

    the study’s findings through triangulation of the data.

    The study findings showed that participation in problem-posing activities can

    facilitate the re-engagement of disengaged, middle-year mathematics

    students. In addition, participation in these activities can result in improved

    problem-solving competence and associated developmental learning

    changes. Some of the changes that were evident as a result of this study

    included improvements in self-regulation, increased integration of prior

    knowledge with new knowledge and increased and contextualised

    socialisation.

  • vi

    TABLE OF CONTENTS

    Certificate ____________________________________________________ ii

    Definition of Acronyms __________________________________________ iii

    Keywords ____________________________________________________ iv

    Abstract ______________________________________________________ v

    List of Tables__________________________________________________ x

    List of Figures ________________________________________________ xii

    List of Appendices ____________________________________________ xiii

    Statement of Authenticity _______________________________________ xiv

    Acknowledgments _____________________________________________ xv

    Chapter 1 - Introduction to the Research Study

    1.1 INTRODUCTION ___________________________________________ 16

    1.2 DEFINITION OF TERMS _____________________________________ 16

    1.3 RATIONALE FOR THE STUDY _________________________________ 18

    1.3.1 Summary ____________________________________________ 25

    1.4 BACKGROUND TO THE STUDY ________________________________ 26

    1.4.1 The Value of Problem Solving in Today’s Society _____________ 27

    1.4.2 The Place of Problem Posing in a Responsive Curriculum ______ 28

    1.4.3 Disparity in Student Mathematical Performance ______________ 31

    1.5 PURPOSE OF THIS STUDY ___________________________________ 32

    1.6 SIGNIFICANCE OF THE RESEARCH _____________________________ 32

    1.7 THESIS OVERVIEW ________________________________________ 34

    Chapter 2 - Theoretical Perspectives

    2.1 CHAPTER OVERVIEW ______________________________________ 36

    2.2 UNDERSTANDING DEVELOPMENTAL LEARNING ____________________ 36

    2.2.1 Information Processing Theory ___________________________ 39

    2.2.2 Psychometric Theory ___________________________________ 42

    2.2.3 Multiple Intelligences Theory _____________________________ 50

    2.2.4 Summary ____________________________________________ 51

    2.3 PROBLEM-SOLVING PERSPECTIVES ____________________________ 53

    2.3.1 Introduction ___________________________________________ 53

    2.3.2 The Power of Teaching through Problem Solving _____________ 56

    2.3.3 Can Problem Solving Drive Mathematical Reform? ___________ 57

  • vii

    2.3.4 Issues Related to the Assessment of PSC __________________ 58

    2.3.5 Should Specific Problem-solving Strategies be Taught? ________ 59

    2.3.6 Student's Understandings of Problem Structures _____________ 60

    2.3.7 Summary ____________________________________________ 63

    2.4 PROBLEM-POSING PERSPECTIVES ____________________________ 65

    2.4.1 Introduction ___________________________________________ 66

    2.4.2 Problem Posing as a Tool for Mathematical Reform ___________ 66

    2.4.3 Problem-posing Skills for Lifelong Learning _________________ 68

    2.4.4 Fostering a Problem-posing Environment ___________________ 70

    2.4.5 Connections between Problem Solving and Problem Posing ____ 71

    2.4.6 Summary ____________________________________________ 75

    2.5 STUDENT UNDERACHIEVEMENT PERSPECTIVES ___________________ 76

    2.6 CONSTRUCTIONIST PERSPECTIVES ____________________________ 80

    2.7 CONCLUSION ____________________________________________ 83

    Chapter 3 - Research Design

    3.1 CHAPTER OVERVIEW ______________________________________ 87

    3.2 INTRODUCTION ___________________________________________ 87

    3.3 RESEARCH QUESTIONS ____________________________________ 92

    3.4 RESEARCH DESIGN _______________________________________ 93

    3.4.1 Research Design Rationale and Structure __________________ 93

    3.4.2 Participants ___________________________________________ 96

    3.5 METHODS _____________________________________________ 102

    3.5.1 Data Collection _______________________________________ 105

    3.5.2 Instruments __________________________________________ 111

    3.5.2.1 The Middle Years Ability Test (MYAT) __________________ 111

    3.5.2.2 The Profiles of Problem Solving (POPS) Test ____________ 113

    3.5.2.3 The Student Survey ________________________________ 115

    3.5.2.4 The Problem Criteria Sheet __________________________ 116

    3.5.3 Data Analysis ________________________________________ 117

    3.5.3.1 Researcher Journal ________________________________ 118

    3.5.3.2 Student Surveys ___________________________________ 118

    3.5.3.3 Student Workbooks ________________________________ 120

    3.5.3.4 Researcher Observations ___________________________ 120

    3.5.3.5 Informal Interviews _________________________________ 121

    3.5.3.6 The Profiles of Problem Solving (POPS) Test ____________ 122

    3.5.4 Reliability and Validity Issues ____________________________ 123

  • viii

    3.5.5 Ethical Issues ________________________________________ 125

    3.6 CONCLUSION ___________________________________________ 127

    Chapter 4 - The Teaching Experiment

    4.1 CHAPTER OVERVIEW _____________________________________ 129

    4.2 THE PHILOSOPHICAL UNDERPINNINGS AND STRUCTURE OF THE

    TEACHING EXPERIMENT____________________________________ 130

    4.2.1 The Philosophical Underpinnings of the Teaching Experiment 130

    4.2.2 The Structure of the Teaching Experiment _________________ 133

    4.3 THE PRE-TEST AND POST-TEST LESSONS ______________________ 134

    4.3.1 Introduction __________________________________________ 134

    4.3.2 First Lesson - Pre-test and Initial Survey ___________________ 136

    4.3.3 Last Lesson - Post-test and Final Survey __________________ 137

    4.4 THE SEVEN TEACHING EPISODES (LESSONS 2-8) ________________ 137

    4.4.1 The First Teaching Episode - Lesson 2 ____________________ 138

    4.4.2 The Second Teaching Episode - Lesson 3 _________________ 139

    4.4.3 The Third Teaching Episode - Lesson 4 ___________________ 143

    4.4.4 The Fourth Teaching Episode - Lesson 5 __________________ 143

    4.4.5 The Fifth Teaching Episode - Lesson 6 ____________________ 144

    4.4.6 The Sixth Teaching Episode - Lesson 7 ___________________ 146

    4.4.7 The Seventh Teaching Episode - Lesson 8 _________________ 148

    4.5 CONCLUSION ___________________________________________ 148

    Chapter 5 - Reporting and Analysis of the Data

    5.1 CHAPTER OVERVIEW _____________________________________ 150

    5.2 OBSERVATIONS AND INTERVIEWS WITH THREE CASE STUDY STUDENTS 150

    5.2.1 Paul _______________________________________________ 152

    5.2.2 Andrew _____________________________________________ 161

    5.2.3 Nicole ______________________________________________ 170

    5.3 STUDENT SURVEYS ______________________________________ 176

    5.3.1 Question One ________________________________________ 176

    5.3.2 Question Two ________________________________________ 179

    5.3.3 Question Three_______________________________________ 182

    5.3.4 Question Four ________________________________________ 184

    5.4 PROFILES OF PROBLEM SOLVING TEST - THE PRE-TEST AND THE

    POST-TEST _____________________________________________ 186

    5.4.1 Descriptive Analysis of the POPS Test Results ______________ 190

    5.4.2 Paired Samples T-Test Results __________________________ 195

  • ix

    5.4.3 Analysis of Improvement of Scores from the Pre-test and the

    Post- test ___________________________________________ 198

    5.5 CONCLUSION ___________________________________________ 201

    Chapter 6 - Responses to the Research Questions

    6.1 CHAPTER OVERVIEW _____________________________________ 204

    6.2 RESEARCH QUESTION 1 ___________________________________ 204

    6.3 RESEARCH QUESTION 2 ___________________________________ 207

    6.4 RESEARCH QUESTION 3 ___________________________________ 209

    6.5 CONCLUSION ___________________________________________ 213

    Chapter 7 - Limitations and Implications for Future Research

    7.1 CHAPTER OVERVIEW _____________________________________ 215

    7.2 LIMITATIONS OF THE STUDY ________________________________ 215

    7.2.1 Limitations in the Selection of Students ____________________ 216

    7.2.2 Limitations in the Timing of the Research __________________ 217

    7.2.3 Limitations of the Size of the Control and Intervention Groups __ 218

    7.2.4 Limitations of the Withdrawal of Students from their Usual

    Classroom Environment ________________________________ 219

    7.2.5 Limitations in the Length of the Problem-posing Intervention ___ 220

    7.2.6 Limitations of Question Three of the Student Survey _________ 222

    7.3 IMPLICATIONS OF THE RESEACH _____________________________ 222

    7.4 CONCLUDING COMMENTS __________________________________ 223

    REFERENCES _____________________________________________ 225

    APPENDICES ______________________________________________ 252

  • x

    L IST OF TABLES

    Table 1.1 Percentage Comparison of How Time is Allocated in

    Year Eight mathematics Classrooms in Germany, the

    United States and Japan

    23

    Table 1.2 Comparative Problem-solving Scale Scores from the

    2003 PISA Test

    24

    Table 1.3 Overall Combined mathematical Literacy Scores from

    the 2003 PISA Test

    25

    Table 1.4 The Eight Skill Groupings of the Employability Skills

    Framework

    29

    Table 2.1 Comparison of Stage Development in Cognitive

    Development Theories

    39

    Table 2.2 Spearman’s Correlations of Student Scores Between

    Subjects

    45

    Table 3.1 Data Used to Respond to the Three Research

    Questions of the Study

    107

    Table 4.1 Variations to Pre-arranged Lesson Times in 2007 135

    Table 5.1 Paul’s Profiles of Problem Solving Pre-test and Post-

    test Results

    153

    Table 5.2 Andrew’s Profiles of Problem Solving Pre-test and

    Post-test Results

    163

    Table 5.3 Nicole’s Profiles of Problem Solving Pre-test and Post-

    test Results

    171

    Table 5.4 Do you enjoy solving problems? 177

    Table 5.5 What type of problems do you prefer to solve? 180

    Table 5.6 Do you think learning to solve problems is a useful

    thing to do?

    183

    Table 5.7 What things could teachers do to assist you to become

    better at solving problems?

    185

    Table 5.8 Comparison Group Pre-test and Post-test results 188

    Table 5.9 Intervention Group Pre-test and Post-test results 189

  • xi

    Table 5.10 Mean Score and Standard Deviation Statistics for each

    Aspect of the Profiles of Problem Solving Test for

    Students in the Comparison and Intervention Groups

    191

    Table 5.11 Paired Samples Test for each Subscale of the Profiles

    of Problem Solving Test for Students in the

    Comparison and Intervention Groups

    197

    Table 5.12 Numbers of Improvements in Individual Aspect Scores

    of Comparison and Intervention Group Students, from

    the Pre-test to the Post-test

    198

  • xii

    List of Figures

    Figure 2.1 A Schematic Diagram of Sternberg’s Triarchic Theory

    of Intelligence

    42

    Figure 3.1 Research Study Framework 95

  • xiii

    L IST OF APPENDICES

    Appendix A Project Information Sheet and Parent Consent Form

    for Comparison Group

    253

    Appendix B Project Information Sheet and Parent Consent Form

    for Intervention Group

    258

    Appendix C Student Survey Sheet 263

    Appendix D Teaching Experiment Lesson One 266

    Appendix E Teaching Experiment Lesson Two 270

    Appendix F Teaching Experiment Lesson Three 275

    Appendix G Teaching Experiment Lesson Four 280

    Appendix H Teaching Experiment Lesson Five 288

    Appendix I Teaching Experiment Lesson Six 295

    Appendix J Teaching Experiment Lesson Seven 302

    Appendix K Teaching Experiment Lesson Eight 309

    Appendix L Teaching Experiment Lesson Nine 314

    Appendix M Profiles of Problem Solving Assessment Instrument

    (Stacey, Groves, Bourke, & Doig, 1993)

    317

    Appendix N Problem Criteria Sheet 327

    Appendix O Participant Pseudonym Code to Psuedonym Name

    Conversion for Comparison Group

    329

    Appendix P Participant Pseudonym Code to Psuedonym Name

    Conversion for Intervention Group

    331

    Appendix Q Marking Scheme for Profiles of Problem Solving Test 333

  • xiv

    STATEMENT OF AUTHENTICITY

    The work contained in this document has not previously been submitted for a

    degree or diploma at any other higher education institution. To the best of my

    knowledge and belief, the document contains no material previously

    published or written by another person except where due reference is made

    in the document itself.

    Deborah Jean Priest

    May 2009

  • 15

    ACKNOW LEDGEMENTS

    I wish to acknowledge the valuable and ongoing support I have received from

    Professor Lyn English in the first instance, and also Dr Mal Shield and Associate

    Professor Rod Nason. In particular, I wish to thank Professor Lyn English and

    Dr Mal Shield for their substantial guidance and encouragement that has been

    instrumental in the completion of my PhD journey. In addition, I would like to

    thank Dr Mark Bahr for his assistance in becoming familiar with the Statistical

    Package for Social Sciences software (SPSS Inc., 2007).

    I would also like to thank the Year 7 teachers, the Deputy Principal in charge of

    the Year 7 students at the research school, and the Principal for allowing me to

    work with their students. I would like to acknowledge you all by name but am

    unable to do so as the participants in this study may be more readily identified as

    a result. Please accept my deepest appreciation for your cooperation and

    assistance.

    As those who have previously completed their PhD journeys will fully understand,

    there are many activities that must be set aside in order to find the necessary

    time to undertake and complete detailed research such as that reported in this

    document. My journey to completion would not have been possible without the

    understanding of my husband, John and my two daughters Megan and Ashley.

    Their patience has been greatly appreciated, cannot be understated and will be

    rewarded in the future.

  • 16

    Chapter 1

    Introduction to the

    Research Study

    1.1 Chapter Overview

    Seven main sections comprise this chapter. The first section is a definition of

    terms used frequently throughout this report (see Section 1.2), while the second

    section introduces the rationale that led to the overarching question for this study

    (see Section 1.3). The third section provides some preliminary background to

    the research study including discussion about the value of problem solving and

    problem posing in a contemporary mathematics curriculum and introduces the

    concept of disparity between a student’s actual mathematical performance and

    their potential performance (see Section 1.4). Section four of this chapter

    introduces the three research questions investigated in this study (see Section

    1.5) while the fifth section considers the significance of this research (see Section

    1.6). The final section presents an overview of the chapters in this report (see

    Section 1.7).

    1.2 Definition of Terms

    The following terms, with their associated meaning, are used frequently

    throughout this report:

    Cognition “refers to the processes or faculties by which knowledge is acquired

    and manipulated” (Bjorklund, 2000, p. 3) and “includes conscious, effortful

    processes such as those involved in making important decisions and

  • 17

    unconscious, automatic processes, such as those involved in recognizing a

    familiar face, word or object” (pp.19, 20).

    Developmental learning changes refer to cognitive (e.g., Goswami, 2002) and

    behavioural (e.g., Lesh & Doerr, 2003) changes that can be attributed to an

    intervention or experiences that occurred over a period of time.

    Engagement refers to the willing participation of students in activities (Ryan &

    Patrick, 2001).

    Middle years refer to Years 5 - 9 in Australian schools. Students enrolled in

    these year levels are most commonly aged between 10 and 14 years.

    Problem posing is the act of creating a new problem for oneself or for peers to

    solve. The problem may be presented in an oral, written or other visual format

    (English, Fox, & Watters, 2005; Lowrie, 2002).

    Problem solving occurs when a specific goal exists that cannot be solved

    immediately due to the presence of one or more obstacles (DeLoache, Miller, &

    Pierroutsakos, 1998). Problem solving is “getting from givens to goals when a

    solution path is not readily apparent” and requires the problem solver to recall

    information, draw upon previously learned skills, choose appropriate solution

    strategies, and express information in a meaningful way. It involves the

    acquisition and utilisation of knowledge, metacognition and socio-cultural

    contexts (Lesh & Zawojewski, 2007).

    Self-regulation refers to a student’s ability to be actively and productively

    involved in an activity that does not intentionally distract or interfere with the

    learning of other students (Schunk, 2001).

    Underachievement occurs when there is a “distance between the actual

    developmental level [of a child] as determined by independent problem solving

    and the level of potential development as determined through problem solving

  • 18

    under adult guidance or in collaboration with more capable peers” (Vygotsky,

    1978, p. 86). In this study, underachieving students will be defined to be those

    students who achieve above average results in the MYAT test (Australian

    Council for Educational Research, 2005) while also achieving lower than the

    average results in the problem-solving criterion of their mathematics tests,

    compared to their cohort.

    1.3 Rationale for the Study

    It could be said that today, children resemble their times more than they

    resemble their parents. This is not surprising when we consider that our current

    times are typified by dynamic advances in technology and a resultant, ever-

    changing job market that requires the workforce to embrace flexibility and

    creativity. The responsibility to prepare our students to be effective and

    productive citizens in such a world is mandated, in part, to the education system

    of the day. In response to this mandate, rigorous reviews of the State-based

    education systems in Australia have lead the Australian Federal Government to

    move towards a national, futures-focussed curriculum that recognises “that

    society will be complex, with workers competing in a global market, needing to

    know how to learn, adapt, create, communicate, and interpret and use

    information critically” (National Curriculum Board, 2008, p.5).

    Two reports the Australian National Numeracy Review Report (National

    Numeracy Review, 2008) and Foundations for Success: The final report of the

    National Mathematics Advisory Panel (National Mathematics Advisory Panel,

    2008) from the United States of America, have provided a foundation for

    discussion papers leading to the development of an Australian, national

    mathematics curriculum. The establishment of this national mathematics

    curriculum is a unique opportunity to redefine not only the appropriate curriculum

    content, but also to reconsider and redefine the most appropriate pedagogy to

    achieve the desired student outcomes.

  • 19

    With mathematics education having a long history of marginalising and

    disengaging students through traditional teaching practices, one could argue that

    a review of teaching practices is timely (English, 2002; Lesh & Zawojewski, 2007;

    Skovsmose & Valero, 2002). Currently not all students are being presented with

    mathematics curriculums that allow them to draw on their knowledge to solve

    meaningful problems that are relevant to them and to society. Indeed, “an

    unintended effect of current classroom practice is to exclude some students from

    future mathematics study” therefore creating a need to engage more students in

    mathematical activities that are connected meaningfully to real-life contexts

    (National Curriculum Board, 2008).

    Education departments and national curriculum organisations across Australia

    have continued to develop policies to promote contemporary teaching practices

    to address this concern, with mixed success. The New Basics Framework is an

    example of a recent four-year project in Queensland schools that created new

    opportunities to connect the curriculum to real-life contexts (Department of

    Education Training and the Arts, 2007). The Framework provided an alternative

    organisational and conceptual framework for the curriculum and was intended to

    reflect the new demands placed on students, and hence on curriculums,

    assessment and pedagogy, by the “new times”.

    The New Basic’s trial curriculum was organised around four “clusters”; Life

    Pathways and Social Futures; Active Citizenship, Multiliteracies and

    Communication Media, and Environments and Technologies. Assessment was

    adapted from assessing and reporting against students’ learning outcomes,

    through traditional pen and paper tests, to student demonstrations of learning

    throughout the transdisciplinary “Rich Tasks”. While some Queensland State

    schools have continued, in part, to pursue and support this new direction in

    curriculum ideology, broad-scale implementation of the Rich Tasks has not

    subsequently occurred across all Queensland schools. Reasons given for the

    lack of broad-scale implementation included insufficient professional

  • 20

    development for teachers and reduced class time available for the development

    of basic student literacy and numeracy skills that will now be measured and

    compared between the States of Australia (Department of Education Training

    and the Arts, 2007). The first national comparison of literacy and numeracy skills

    between students in different States took place in May 2008 as part of the

    National Assessment Program Literacy and Numeracy (NAPLAN) (Ministerial

    Council on Education, Employment, Training, & Affairs, 2008a).

    Despite the Queensland Government’s decision not to proceed with the full

    implementation of the New Basics Framework, curriculum organisations are still

    calling for meaningful connections to be made between school-based

    curriculums and real-life contexts (Department of Education Training and the

    Arts, 2007). In 2008, the National Curriculum Board of Australia opened public

    discussion about what is important in the teaching of mathematics, by publishing

    for public comment, papers about a nationally administered mathematics

    curriculum. One such paper, The National Mathematics Curriculum: Framing

    Paper argued that “mathematics is important for all citizens” and that “some

    students are currently excluded from effective mathematics study” (National

    Curriculum Board, 2008, p. 1). The paper stated that equity of opportunity is a

    central goal in the construction of a national mathematics curriculum and

    included discussion about how specific groups have been excluded and how to

    re-engage more students in the study of mathematics. According to the paper,

    the students at most risk of disengagement are students in their middle years of

    schooling. The paper suggested the alienation and disengagement of these

    students is largely attributed to “irrelevant curriculums”, unconnected to real-life

    contexts, and “ineffectual learning and teaching processes” (National Curriculum

    Board, 2008, p. 5). The report went on to state that it is “imperative that we

    reverse this trend” (p. 5).

    The concept of irrelevant curriculums is not new (Secada & Berman, 1999) with

    Hollingsworth, Lokan and McCrae (2003) reporting that, in Year 8 mathematics

  • 21

    lessons, more than seventy-five percent of the problems provided to students

    were low in complexity, emphasised procedural fluency, rather than higher-order

    critical thinking, and only twenty-five percent of the problems were connected to

    real-life contexts. When looking a little further into the senior years of schooling,

    Barrington (2006) reported a drop in student participation rates in Year 12

    mathematics classes and the National Numeracy Review (NNR) reported a

    decline in tertiary students undertaking substantial studies in mathematics which,

    in part, has lead to a national shortage of secondary mathematics teachers

    (National Numeracy Review, 2008). These reports have major implications for

    educators and in particular, teachers of middle-year mathematics; for it is in the

    middle years of schooling that students appear to be forming enduring

    dispositions and perceptions about the personal relevance of the study of

    mathematics that can lead to underachievement, disengagement or both

    (National Curriculum Board, 2008).

    According to the National Declaration on Education Goals for Young Australians

    draft report, Australia has no “inherent advantage – except through the quality of

    education” to prepare students for the “radically evolving and uncertain context”

    of future life in a global society (Ministerial Council on Education, Employment,

    Training, & Affairs, 2008b, p. 4). This is supported in the National Mathematics

    Curriculum: Framing paper where the authors stated that “a fundamental goal of

    the mathematics curriculum is to educate students to be active, thinking citizens,

    interpreting the world mathematically, and using mathematics to help form their

    predictions and decisions about personal and financial priorities” (National

    Curriculum Board, 2008, p. 3).

    The paper defined, as goals of a national mathematics curriculum, four

    proficiency strands;

    1. understanding (conceptual understanding);

    2. fluency (procedural fluency);

  • 22

    3. problem solving (strategic competence) and,

    4. reasoning (adaptive reasoning).

    It stated that problem-solving competence, including “the ability to make choices,

    interpret, formulate, model and investigate problem situations, and communicate

    solutions effectively”, is central to ensuring a futures orientation to a national

    curriculum (National Curriculum Board, 2008, p. 3). The importance of

    developing problem-solving competence was previously discussed by Cai (2003)

    during his investigation of Singaporean students’ mathematical thinking in

    problem solving and problem posing. Cai stated, following his exploratory study,

    that problem solving was the most purposeful activity in the study of

    mathematics. Later researchers such as Brown and Walter (2005) suggested

    that it was the formulation or posing of problems, more so than the solving of

    problems, that was fundamental in the development of mathematical skills.

    Previous researchers such as Lowrie (2002) had already undertaken some

    research into the usefulness of problem posing and had discovered that, when

    used as a regular strategy in the study of mathematics, problem posing had the

    potential to increase the engagement of underachieving students.

    Shimizu (2002) also considered the engagement of students when he

    investigated how the structured problem-solving approach to teaching

    mathematics in Japanese schools and its associated impact on how Japanese

    students perceive their lessons, compared with the pedagogy used by German

    and American mathematics teachers and the perceptions of their students. One

    of the differences he noted was that fostering mathematical thinking was the

    main goal of mathematics lessons for the majority of Japanese teachers whereas

    61 percent of American teachers and 55 percent of German teachers had the

    development of mathematical skills as their main goal. A second difference he

    discussed was the time spent by Japanese, German and American students on

    the practice of routine procedures compared to time spent thinking about

  • 23

    mathematical problems and inventing new solutions. His data were taken from a

    Third International Mathematics and Science Study (TIMSS) video classroom

    study (Stigler, Gonzales, Kawanaka, Knoll, & Serrano, 1999) and can be found in

    the Table 1.1.

    Table 1.1

    Percentage Comparison of How Time is Allocated in Year Eight Mathematics

    Classrooms in Germany, the United States and Japan

    OECD Country Practising routine

    procedures

    Thinking about

    mathematical problems and

    inventing new solutions

    Japan 40.8 44.1

    Germany 89.4 4.3

    United States 95.8 0.7

    Note. Adapted from " The TIMSS videotape classroom study: Methods and findings from an exploratory research project on eighth-grade mathematics instruction in Germany, Japan, and the United States” by J.W. Stigler, P. Gonzales, T. Kawanaka, S. Knoll & A. Serrano, 1999, Washington, D.C..

    According to the study (Stigler et al., 1999), students in Japan spend less than

    half the amount of time practising routine procedures and more than ten times

    the amount of time working with problems than do their German and American

    counterparts. These statistics become more notable when we consider the

    performance of Japanese students compared to American and German students

    in The Program for International Student Assessment (PISA) test undertaken by

    students in twenty-nine member countries of the Organization for Economic

  • 24

    Cooperation and Development (OECD) in 2003 (Lemke et al., 2004). The

    average country scores for fifteen-year-old students from Japan, Germany,

    Australia and the United States on the problem-solving scale are reported in

    Table 1.2 while the overall combined mathematical literacy scores can be found

    in Table 1.3. Statistics about Australian students have been included for

    comparative purposes.

    Table 1.2

    Comparative Problem-solving Scale Scores from the 2003 PISA Test

    OECD Country Average student score

    (average = 500, S.D.=100)

    OECD Ranking

    N=29

    Japan 547 3rd

    Australia 530 5th

    Germany 513 13th

    United States 477 24th

    Note. Adapted from "International outcomes of learning in mathematics, literacy and problem solving: PISA 2003 results from the U.S. perspective - highlights” by M. Lemke, A. Sen, E. Pahlke, L. Partelow, D. Miller & T. Williams, 2004, Washington, D.C.: National Center for Educational Statistics.

    It could be deduced from the results in Tables 1.1, 1.2 and 1.3 that a

    mathematics classroom rich in problem-solving opportunities can not only lead to

    enhanced performance on international problem-solving testing instruments, it

    can also support the development of mathematical literacy and is therefore

    worthy of further research in an Australian school context.

  • 25

    Table 1.3

    Overall Combined Mathematical Literacy Scores from the 2003 PISA Test

    OECD Country Average student score

    (average = 500, S.D.=100)

    OECD Ranking

    N=29

    Japan 534 4th

    Australia 524 8th

    Germany 503 16th

    United States 483 24th

    Note. Adapted from "International outcomes of learning in mathematics, literacy and problem solving: PISA 2003 results from the U.S. perspective - highlights” by M. Lemke, A. Sen, E. Pahlke, L. Partelow, D. Miller & T. Williams, 2004, Washington, D.C.: National Center for Educational Statistics.

    1.3.1 Summary

    Four foci arose from the preliminary review of the literature surrounding

    mathematics education in Australia and internationally:

    1. problem solving

    2. problem posing

    3. middle years and,

    4. underachievement

    It has been suggested that posing problems can re-engage underachieving

    students (Lowrie, 2002) and that middle-year students are at most risk of being

  • 26

    disengaged and underachieving in the study of mathematics (National

    Curriculum Board, 2008). Problem posing has been attributed as being an

    important skill in the development of problem-solving competence (e.g., Cai,

    2003; English, 2003; Silver & Cai, 1993a), which is one of the four proficiency

    strands that make up the structure of the new national mathematics curriculum

    (National Curriculum Board, 2008). International research has reinforced the

    value of a mathematics curriculum, rich in problem-solving, to the student

    development of mathematical literacy skills and problem-solving competence

    (Lemke et al., 2004; Stigler et al., 1999). In light of these observations and to

    progress the reform of mathematical curriculums, the following overarching

    question was investigated in this present research study:

    How might a problem-posing intervention impact upon the development of

    problem-solving competence of underachieving, middle-year students?

    The decision to investigate this overarching research question was consistent

    with international curriculum documents such as those written by the American

    National Research Council (e.g., NRC, 2004), and the National Council of

    Teachers of Mathematics (NCTM, 2000) that recommended that teachers

    provide regular opportunities for students to pose and solve problems within

    meaningful contexts. The results of this present study provide education policy

    makers, syllabus writers, and teachers with insights into how underachieving,

    middle-year, mathematics students may be assisted to develop problem-solving

    competence through a problem-posing intervention (e.g., Bjorklund, 2000; Jones

    & Myhill, 2004; Kanevsky & Keighley, 2003).

    1.4 Background to the Study

    This section considers further background information on problem solving,

    problem posing and disparity between student results and their potential that was

    used to develop the three Research Questions for this study.

  • 27

    1.4.1 The Value of Problem Solving in Today’s Society

    Problem solving is widely argued as the most purposeful activity in a

    mathematics curriculum (Cai, 2003; Cai & Hwang, 2002; Costa, 2005; NCTM,

    2000). It is not surprising then to find the States of Australia have been collecting

    data about students’ problem-solving performance from all students in Years 3, 5

    and 7 for almost ten years (e.g., Queensland Studies Authority, 2005). Despite

    these and similar efforts at collecting data, it seems that little of the data have

    been converted into reform of the teaching and learning of mathematics (e.g.,

    Lowrie, 2002). As international researchers (e.g., Brown & Walter, 2005; Lester,

    2003) have indicated, a review of current practices was needed, as was a “fresh

    perspective of problem solving … that goes beyond current school curricula and

    state standards” (Lesh & Zawojewski, 2007, p. 52). Of equal concern is Lesh

    and Zawojewski’s recent review of literature that reported there is a “growing

    recognition that a serious mismatch exists (and is growing) between the low-level

    skills emphasized in test-driven curriculum materials and the kind of

    understanding and abilities that are needed for success beyond school”

    (Gainsburg 2003a in Lesh & Zawojewski, 2007 pp. 5-6). In fact, they went so far

    as to say that the challenging and novel problems encountered outside of the

    school environment, requiring extensive use of mathematics, are frequently

    inconsistent with the underlying assumptions of conventional approaches to

    solving mathematical problems in schools (Lesh & Zawojewski, 2007).

    Indeed, the extent to which our education system is successful in developing

    these skills has broad implications for students as they leave the school system.

    Universities and employers throughout Australia and overseas are looking to

    organisations like the Australian Council for Educational Research (ACER) to

    screen prospective students and employees for their problem-solving

    intelligence. Testing instruments such as the Commonwealth Government

    funded Graduate Skills Assessment (GSA) (ACER, 2003), can now be used by

  • 28

    employers and universities to assist in the determination of university placements

    and employment suitability.

    Further evidence for the value of problem solving in Australian society can be

    found in a more recent government initiative, which saw the Department of

    Education, Science and Training (DEST) and the Australian National Training

    Authority (ANTA) contract a project to establish the Employability Skills

    Framework (DEST & ANTA, 2004). The purpose of this project was to inform

    educators about employer perspectives on the personal attributes and skills of

    desirable employees. The framework specified eight skill groupings that defined

    and described employability skills (see Table 1.4).

    There is a need to “continue building Australia’s capacity to effectively operate in

    the global knowledge-based economy” and “education and training providers will

    have a key role in equipping the community for this challenge” (Australian

    Chamber of Commerce and Industry, 2002, p. 1). Reports, such as the

    Employability Skills Framework (DEST & ANTA, 2004) attempt to address this

    need and provide implications for researchers of educational pedagogy. Not only

    is the acquisition of problem-solving competence fundamental in acquiring

    important mathematical concepts (e.g., Adams, Brower, Hill, & Marshall, 2000;

    Bobis, Mulligan, & Lowrie, 2004), it can also impact on the employability of

    graduates entering the work place.

    1.4.2 The Place of Problem Posing in a Responsive Curriculum

    Problem-posing skills are a fundamental building block in the development of

    mathematical skills (Brown & Walter, 2005; Lowrie, 2002; NCTM, 2000).

    Problem-posing activities are a means to demystify problems and to empower

    students to connect with mathematics in a more personal and meaningful way.

    However, despite the clear benefits of problem-posing activities, students are not

    often given the opportunity to pose their own mathematics problems publicly

    (Silver, 1997).

  • 29

    Table 1.4

    The Eight Skill Groupings of the Employability Skills Framework

    Skill Description

    Communication Skills that contribute to productive and harmonious relationships between employees and customers

    Team Work Skills that contribute to productive working relationships and outcomes

    Problem-solving Skills that contribute to productive outcomes

    Initiative and enterprise Skills that contribute to innovative outcomes

    Planning and organisation Skills that contribute to long-term and short-term strategic planning

    Self-management Skills that contribute to employee satisfaction and growth

    Learning Skills that contribute to ongoing improvement and expansion in employee and company operations and outcomes

    Technology Skills that contribute to effective execution of tasks

    Note. From “Employability skills final report: Development of a strategy to support universal recognition and recording of employability skills - A skills portfolio approach.” by Department of Education, Science and Technology and Australian National Training Authority. 2004. Canberra, ACT.

    The virtues and benefits to students of posing problems have been known for

    some time. Hart (1981) marvelled at how the activity of allowing students to pose

  • 30

    their own problems afforded her the opportunity to “open a window” through

    which to view students’ thinking. Van Den Brink (1987) expressed a similar view

    when he said problem posing provided him with a “mirror” that reflected the

    content and character of a student’s mathematical experience. However, Silver

    and Cai (1993b) suggested more profound reasons for including problem posing

    as a learning activity, as it presents the opportunity to consider students’ views

    on issues of morality, justice and human relationships. These virtues and

    benefits are as valid today as they were twenty years ago.

    Research has been undertaken in recent years that also espouses the benefits of

    mathematical problem posing and solving in a balanced mathematics curriculum

    (Bjorklund, 2000; Bobis et al., 2004; Brown & Walter, 2005; Cai, 2003; Daniel,

    2003; English et al., 2005; Knuth & Peterson, 2002; Stoyanova, 2003). While

    problem posing and problem solving feature highly in most Australian and

    American policy documents on Mathematics education, in some American

    mathematics classrooms the learning of knowledge and processes received over

    one hundred times the attention afforded to the development of problem solving

    (Stigler et al., 1999).

    To address the research that suggests traditional practices in the teaching of

    mathematics can contribute to the disengagement of students (e.g., English,

    2002; Lesh & Zawojewski, 2007), research into problem posing has continued

    (e.g., Brown & Walter, 2005; English et al., 2005). Of particular interest are the

    reports by researchers of increased engagement of underachieving students in

    the study of mathematics when problem posing was used as a regular teaching

    strategy (English, 1997a, 1997b; Lowrie, 2002). However, despite these

    findings, connections between a problem-posing intervention and increased

    problem-solving competence of students, who achieve above average results in

    standardised intelligence tests and who underachieve on problem-solving tests,

    are yet to be made. This study has attempted to fill this void in the research.

  • 31

    1.4.3 Disparity in Student Mathematical Performance

    It is widely accepted that the mathematical abilities of students of different ages

    vary enormously; but so do the intellectual abilities of same-aged students (Case,

    1998). These differences have been the study of many research projects

    investigating intelligence and the means to measure intelligence (e.g., Sternberg,

    2002; Vernon, Wickett, Bazana, & Stelmack, 2000). A number of standardised

    intelligence tests have been devised over the past one hundred years and have

    been used to benchmark cognitive development (e.g., Spearman, 1904;

    Wechsler, 1991). These tests distinguish between the mental age of a child and

    the chronological age of a child. The power of the message sent to students

    when their performance on such tests is alluded to, or even articulated to the

    child, cannot be underestimated. What students believe about their intelligence

    and mathematical performance has been shown to be a powerful indicator of

    achievement outcomes (Stipeck & Gralinski, 1996).

    While we can readily accept that mathematical abilities of students vary from

    student to student, it is perplexing when intelligence tests suggest a strong

    potential for mathematical ability, yet results from classroom tests do not support

    this prediction. In particular, the scenario becomes more perplexing when a

    student achieves a high predictive score in an intelligence test, scores highly in

    routine procedural questions in class tests, yet continues to perform below

    average in questions that require significant problem-solving capabilities. This is

    an area of research that has received little attention in the corpus of knowledge

    connecting students and their problem-solving capabilities, and precipitated one

    of the foci of this study.

    1.5 The Purpose of this Present Study

    The purpose of this present study was to investigate how a problem-posing

    intervention might impact on the development of students’ problem-solving

    competence, with a particular focus on the engagement of under-achieving,

    middle-year students. This present study provided opportunities for selected

  • 32

    students, from four different Year 7 classes in the one school, to pose and

    explore their own problems over a seven-lesson teaching experiment. Eighteen

    participants met the selection process (see Section 3.3.2) and were withdrawn

    from their customary Monday morning assembly each week. They met together

    as a group in a multi-purpose classroom in their School library. Data from three

    of these students was disregarded, due to the multiple absences of these

    students from the teaching episodes, leaving data from fifteen students to be

    analysed. From the remaining students, three case-study students were chosen

    for a detailed investigation of the changes that occurred for them as a result of

    the problem-posing intervention (see Section 5.2 for the selection process of the

    three case-study students).

    To address the purpose of this present study, three research questions were

    investigated during the teaching experiment.

    Research Question 1

    Can, and if so, how can participation in problem-posing activities facilitate the re-

    engagement of middle-year mathematics students?

    Research Question 2

    Can, and if so, how can participation in problem-posing activities facilitate

    improved problem-solving competence of middle-year, mathematics students?

    Research Question 3

    In terms of problem-solving competence, what developmental learning changes

    occur during the course of a problem-posing intervention?

    1.6 Significance of the Research

    Ceci (1996) argued that it is not possible to deduce the intelligence of a person

    from their performance on a set of standardised questions such as those found

  • 33

    on commonly used Intelligence Quotient (IQ) tests. Indeed, he argued that

    cognition occurs within the framework defined by parents, teachers, peers, and

    the culture of the time. It follows then that it may not be possible to accurately

    deduce students’ mathematical potential from a set of questions presented to

    them in a standardised test or examination, as is the current status quo in many

    schools across Australia. It has been mooted by several authors that alternative

    activities, such as problem posing, may provide teachers with more authentic and

    accurate insights into their students’ understandings of mathematical processes

    and concepts. Performance at problem-posing tasks may therefore be a more

    accurate indicator of student’s mathematical potential (Anderson, 1997; Bobis et

    al., 2004; Brown & Walter, 2005).

    Siegler (1996) maintained that teachers can influence their students’ cognitive

    development in three significant ways. Firstly, they can influence what their

    students think about. Secondly, they can influence how their students will

    acquire and construct their information and, thirdly, they can influence why their

    students engage in the learning process. This view is supported by Tate and

    Rousseau (2002) who found that mathematics was the favourite subject of most

    Year 1 and 2 students, yet was one of the least favourite by the time they

    reached the middle years of schooling. They attributed this phenomenon to

    either the students removing themselves from the challenging programs in

    mathematics or the teachers removing the challenging programs from them. In

    either situation, mathematics teachers clearly have an important role to play in

    constructing effective learning opportunities for their students.

    The use of a problem-posing intervention has been investigated by many

    researchers. For example, Bandura (1997) discussed the impact of problem-

    posing opportunities on students’ self-efficacy, while Knuth (2002) considered its

    impact on the development of students’ intrinsic motivation to engage in the

    learning of mathematics. Graham, Harris and Larsen (2001) looked at how

    problem posing could be used in the prevention of writing problems for students

  • 34

    with learning difficulties, while Lowrie (2002) focussed on the influence of the

    teacher on the types of problems students pose. Contreras (2003) and Lavy and

    Bershadsky (2003) investigated a problem-posing approach to solving geometry

    problems, while Stoyanova (2003) considered the impact of problem posing on

    gifted and talented students. Despite this apparent breadth of problem-posing

    research, there appears to be little research into the role of a problem-posing

    intervention in assisting underachieving mathematics students who have above-

    average performance in standardised intelligence tests. This study has

    addressed this shortcoming.

    1.7 Thesis Overview

    This thesis comprises seven chapters. The first chapter provides an introduction

    to the research study while the second chapter provides a report on the relevant

    literature pertaining to problem-solving, problem-posing and underachievement

    of students in their middle years of schooling. This review highlights where the

    shortcomings in the research exist. A discussion about the design and

    theoretical foundations of the research study and a detailed description of the

    instruments used to collect data, can be found in Chapter Three. This chapter

    also includes a section outlining the selection process for participants of this

    study and a more detailed description of how three case-study students came to

    be chosen from the participant group.

    Issues pertaining to reliability and validity of the data collected and the

    associated ethical considerations arising from this study are discussed towards

    the end of Chapter Three. Chapter Four introduces the structure of the teaching

    experiment and discusses each teaching episode in detail. These discussions

    are particularly useful in highlighting the situational challenges, and associated

    implications for data collection and analysis, that arose throughout the

    experiment. The fifth chapter reports on the data collected during the teaching

    experiment and contains an in-depth review of the impact of the problem-posing

    intervention on three case-study students; Paul, Andrew and Nicole. Chapter Six

  • 35

    provides an analysis and synthesis of the data collected throughout the teaching

    experiment that enabled the three Research Questions to be answered. The

    limitations of this study and the implications of the study’s findings for future

    research are discussed in Chapter Seven.

  • 36

    Chapter 2

    Theoretical Perspectives

    2.1 Chapter Overview

    This chapter contains a critical review of current literature pertaining to this

    present study. The review begins in Section 2.2 with the literature pertaining to

    the developmental learning of students. It starts with a brief introduction to the

    main theories discussed by education researchers and then focuses on the three

    theories that are particularly relevant to educational research related to the

    learning of mathematics. The literature surrounding the development of problem-

    solving competence and its relevance and role in developing mathematical skills

    is reviewed in Section 2.3, while literature about the use of problem-posing as an

    intervention to promote student learning is reviewed in Section 2.4. This latter

    section concludes with a review of the literature surrounding the relationship

    between the development of problem-solving competence and student

    opportunities to pose their own problems. Literature related to the possible

    causes of underachievement of middle-year students is reviewed in Section 2.5.

    The literature surrounding the theoretical framework that underpins this present

    study and the investigation of the Research Questions is presented in Section

    2.6, while a conclusion for the chapter can be found in Section 2.7.

    2.2 Understanding Developmental Learning

    “Developing an understanding of the developmental status of students’ thinking

    and learning is fundamental to improving that learning” (Cai & Hwang, 2002, p.

    401). As student development of problem-solving competence was a goal of this

    present study, this section presents an overview of the literature surrounding

    developmental learning of students. Links between developmental learning and

  • 37

    problem-solving competence are established, as are the areas in the research

    where disagreement between authors exists and uncertainty occurs.

    Researchers have provided many methods and concepts that increase our ability

    to observe, explain and describe the process of student’s developmental

    learning. For example, Siegler (1991) said,

    all types of thinking involve both products and processes. The products

    of thinking are the observable end states – what children know at

    different points in development. The processes of thinking are the

    initial and intermediate steps, often accomplished entirely inside

    people’s heads that produce the products. (p. 3)

    He compared children to scientists because they both ask innumerable

    elementary questions about the nature of the universe, which seem entirely trivial

    to everyone else, and are both given the time by society to pursue their

    ruminations. This inquisitive nature of children is the very attribute that lends

    itself to the development of problem-solving competence and problem-posing

    expertise from a very early age. Siegler (1991) exemplified this view when he

    talked about it not being uncommon to see a toddler in a high chair deliberately

    drop food from their tray onto the floor to see what happened to the food.

    Together with investigations on intelligence and developmental learning,

    researchers are gaining a clearer picture of how to assist students to narrow their

    “zone of proximal development” (Vygotsky, 1978) in problem-solving

    competence. However, it is not clear, from the current research, whether

    problem posing is an appropriate teaching strategy for the particular group of

    middle-year students who underachieve in problem solving, yet who appear to

    have above average intelligence compared to their peers. Whether intelligence

    and developmental learning are a function of nature or nurture has been actively

  • 38

    debated for many years. In fact, many researchers have published a plethora of

    theories, to understand differences in children’s cognition and developmental

    learning, that are worthy of review (e.g., Bjorklund, 2000). Despite some

    researchers supporting conceptual frameworks of more than one theory, for

    example, Sternberg (1999a; 2002) supporting the multiple intelligences and

    information processing theories, and Case (1998) supporting the stage and

    information processing theories, in general, most researchers’ work aligns with

    one of five theories, which are highlighted in Table 2.1. This present study draws

    most heavily from the Information Processing Theory, with some reference made

    to the Multiple Intelligences Theory, and the Psychometric Testing Theory, where

    relevant.

    Regardless of which theory a researcher supports, it is helpful to acknowledge

    three basic characteristics of developmental learning. Firstly, we can

    acknowledge that the brain is capable of finite information storage and

    information processing capacity. Secondly, the human brain is constantly

    adapting to a changing environment and thirdly, Goswami (2002) would have us

    believe that “cognitive skills almost always can be increased, at least to some

    degree” (p. 619). These three characteristics will be discussed further, within the

    context of the Information Processing Theory, the Multiple Intelligences Theory

    and the Psychometric Testing Theory in the next three sections.

  • 39

    Table 2.1

    Comparison of Stage Development in Cognitive Development Theories

    Theory Underpinning Beliefs Leading Researchers

    Stage Learning occurs in stages and a child needs to pass through one stage completely before entering the next stage.

    (Piaget & Inhelder, 1969); (Case, 1998)

    Information Processing

    Mental representations, processes, strategies, and knowledge develop over time.

    (Sternberg, 2002); (Halford, 2002); (Klahr, 1992); (Deary, 2000); (Lohman, 2000); (Siegler, 1991, 1996)

    Psychometric testing

    Intelligence can be described in terms of mental factors and psychometric testing instruments can be constructed to reveal such factors.

    (Spearman, 1904); (Brand, 1996); (Hernstein & Murray, 1994) ; (Jensen, 1998); (Wechsler, 1991)

    Multiple Intelligences

    Intelligence is not a unitary concept, but more a multiple one, where intelligence may be domain specific or domain general.

    (Gardner, 1999a); (Sternberg, 1997a); (Thelan & Smith, 1998);

    Biological, Environmental and Social Factors

    Intelligence characteristics are acquired partly through heredity. Cognitive development occurs through the internalisation of concepts experienced through environmental and social contact.

    (Vygotsky, 1981); (Feuerstein, 1979); (Rogoff, 1998); (Ceci, 1990); (Grigorenko, 2000); (Vernon et al., 2000)

    2.2.1 Information Processing Theory

    Information processing theorists argue that thinking is like processing

    information. The quality of the thinking is dependent on the processing capability

    and memory limitations of the child. In other words, what information the child

  • 40

    chooses to use in a particular situation, how the child processes the information

    to achieve their desired outcome, and how much of the information they can

    retain in memory at anyone time, will be decisive factors in their overall success

    at solving problems. Siegler (1996) spoke about an “essential tension” (p. 58)

    that exists for children between their limitations to retain and process information

    and their automatic striving to find ways to overcome these limitations. He

    discussed a variety of strategies commonly used by children in this pursuit which

    included:

    1. practice and rehearsal to overcome limited memory capacity,

    2. increased use of resources such as books or the internet to overcome

    limited knowledge, and

    3. the use of problem-solving strategies, such as breaking a problem into

    smaller sub-problems, to overcome an inability to deal with long

    sequences of tasks.

    .

    According to Siegler (1991) “it is no accident … that the two main theoretical

    approaches to cognitive development – the Piagetian and the information

    processing approaches – both place great emphasis on problem solving” (p.

    252). He said that when children regularly solve problems they are in fact

    contributing to their own cognitive development as problem solving requires them

    to create solutions for themselves, rather than relying on procedures and

    practised routines they have learnt. This active involvement by a child in their

    own developmental learning, by engaging in continuous self-modification

    (Siegler, 1996), was also supported by Bjorklund (2000) who said “cognitive

    development is a constructive process, with children playing an active role in the

    construction of their own minds” (p. 481).

    Researchers who support an information-processing theory, discuss four change

    mechanisms that they believe play a significant role in childhood cognitive

  • 41

    development: automatisation (the increasingly efficient execution of mental

    processes), encoding (the selection and prioritising of important aspects of

    situations), generalisation (the use of prior knowledge of numerous familiar

    situations), and strategy construction (the synthesis of change processes to

    produce cognitive growth) (see Sternberg, 2000). In previous research

    Sternberg (1985) referred to only three information processing components of

    general intelligence in his Triarchic Theory of Intelligence (see Figure 2.2), these

    being knowledge acquisition components (discrimination between relevant

    and irrelevant data), metacomponents (selection and planning of appropriate

    strategies) and performance components (combination of the selected data

    and appropriate strategy to solve the problem). However, none of these

    components explicitly acknowledge the efficiency with which a student solves a

    problem as a significant factor of intelligence. The efficiency of execution in the

    solution of a problem warrants further investigations where the time allowed for

    an assessment of skills is a controlled factor.

    While Sternberg’s earlier work is over twenty years old, and has been

    superseded by the four change mechanisms to a large extent, researchers (e.g.,

    Goswami, 2002; Thomas & Karmiloff-Smith, 2002) still refer to Sternberg’s

    Triarchic Theory of Intelligence when discussing childhood cognitive

    development. According to Goswami (2002), “individual differences in cognition

    derive largely from individual differences in the execution of these three kinds of

    components. The components are highly interdependent.” (p. 608)

  • 42

    Selective Selective Strategy Strategy Encoding Application Encoding Combination Construction Selection Selective Strategy Inference Comparison Coordination

    Figure 2.1. A Schematic Diagram of Sternberg’s Triarchic Theory of Intelligence

    (in Siegler, 1991 p. 69).

    2.2.2 Psychometric Theory

    If we assume that infants come into the world poorly endowed, the

    question becomes how they are able to develop as rapidly as they do. But

    if we assume that infants come into the world richly endowed, the question

    becomes why development takes so long. (Siegler, 1991, p. 3)

    This section explores the issues surrounding this nature versus nurture debate

    that begun in the late 1800s by researchers such as Sir Francis Galton (1883),

    Charles Darwin’s cousin, who popularised the now famous Bell Curve and its

    associated normal (Gaussian) distribution. A review of the history of

    psychometric theory is relevant to current research as views held by

    contemporary proponents of the psychometric theories have not changed

    Intelligence

    Metacomponents Knowledge Acquisition Components

    Performance Components

  • 43

    significantly to those of the founding researchers in this field. The Bell Curve,

    first introduced by Galton, is still used as a standard tool for comparing students

    and prospective employees as well as being used by researchers for interpreting

    data in social science research projects.

    Galton’s (1883) interest in comparing individuals stemmed from his advocacy for

    eugenics, the inter-breeding of intelligent people in order to strengthen the gene

    pool of the human species. While Galton initially investigated the distribution of

    physical measurements such as weight and height, he later theorised that since

    psychological characteristics were based on physiological characteristics, then

    human intelligence could also be represented by the Bell Curve. While Galton

    had begun founding research into human intelligence, he did not construct broad-

    scale instruments to measure intelligence levels of children or adults. This work

    was taken up a few years later in France when universal education was

    introduced in the late 1890s, as a result of the Industrial Revolution, with

    psychologists Alfred Binet, Director of the Sorbonne in France, and Theophile

    Simon being engaged to develop a testing instrument to determine which

    children needed “special education” (Binet & Simon, 1905).

    The first Binet-Simon test (Binet & Simon, 1905) was used in 1905 and included

    thirty questions on reasoning, memory, language and problem solving, ordered

    by difficulty, and was used to identify children who may experience difficulty with

    a common curriculum. The test was based on data from 50 subjects, therefore

    lacking validity, and was criticised because it relied heavily on the reading and

    language ability of the children. Almost one hundred years later, this same

    criticism is leveled at authors of psychometric tests in current use (Bjorklund,

    2000; Gardner, 1999b). The Binet-Simon test was revised in 1908 following

    further research with 203 subjects and had test items grouped according to age

    level rather than increasing difficulty. It was at this stage that Binet and Simon

    introduced the concept of mental age (MA), as compared to chronological age

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    (CA), which later resulted in the establishment of the ‘intelligence quotient’ (IQ)

    by German psychologist William Stern (1912) and Terman (1916) that is still

    used today to define, label and categorise students.

    At around the same time as Binet and Simon (1905) established their first test,

    research into human intelligence and developmental learning took a different

    direction in England with Spearman (1904) investigating the existence of a

    general intelligence factor that he abbreviated to a more commonly used

    expression, a g factor. According to Spearman, all individual differences in

    cognitive ability were due to a general factor that is present at birth and that he

    believed was as a result of differences in mental energy. This g factor impacted

    upon performance in all cognitive tests, whereas a specific factor, (commonly

    called an s factor) could impact upon an individual’s performance in a specific

    type of test. To support his proposition, he examined correlations between

    student scores on different school subjects (see Table 2.2) and offered the high

    positive correlations as evidence of the existence of a single common general

    intelligence factor. This suggestion of individuals having specific s factors

    maintained momentum and, 85 years later, was paralleled by Ceci’s (1990) view

    that the context in which a test occurs is a decisive and determining agent in an

    individual’s performance on the test. This position has important implications for

    current research where researchers are interested in the participant’s

    developmental learning changes as opposed to their connectedness to the

    context of the questions used in the assessment instrument or the style of the

    questions.

    If the position of specific and general factors of human intelligence was to be

    accepted, a new testing instrument was needed to measure and compare the

    intelligence of individuals. Spearman (1904) in association with Cyril Burt,

    another British psychologist, were some of the earliest researchers to develop a

    range of intelligence tests, to measure the mental abilities of British school

  • 45

    children, that took general and specific intelligence factors into consideration.

    They pioneered the concept of factor analysis that other researchers, such as

    Thurstone (1938) and Wechsler (1991), further developed many years later.

    These tests allowed gender differences to be considered. For example, Halpern

    (1997) reported that, on average, boys score higher on tasks that involve visual

    and spatial awareness than do girls, while girls perform better than boys at tasks

    that require access to long-term memory, fine motor skills, perceptual speed, and

    writing and comprehension of complex prose. These findings require current

    researchers to consider whether assessment instruments favour the natural

    differences of either gender. Without these consderations, the validity of data

    could be challenged.

    Table 2.2.

    Spearman’s Correlations of Student Scores between Subjects

    Subject Classics French English Math Pitch Music

    Classics - .83 .78 .70 .66 .63

    French .83 - .67 .67 .65 .57

    English .78 .67 - .64 .54 .51

    Math .70 .67 .64 - .45 .51

    Pitch .66 .65 .54 .45 - .40

    Music .63 .57 .51 .51 .40 -

    Note. From "General intelligence, objectively determined and measured” by C. Spearman, 1904, American Journal of Psychology, 15(2), pp. 201-293.

  • 46

    By the early 1920s, the use of psychometric testing had expanded to the United

    States and was being used as a means to determine which immigrants were

    suitable for residency and which should be deported, and later in the 1930s to

    determine intelligence levels of American school children. To achieve this goal,

    Lewis Terman (1916), a Stanford Professor, revised the French Binet-Simon

    (1905) test calling it the Stanford-Binet test. Results from this test were

    compared to a standardised sample of 3184 mainly white, urban children from

    eleven states in America, chosen by father’s occupation. This revised test was

    administered under the assumption that not all children of a particular age think

    and reason in the same way or at the same level. Terman’s results were more

    reliable for older children aged between twelve and sixteen years than for

    younger children, or children in the lower IQ ranges, but he found standard

    deviations for children in different age groupings made the interpretation of data

    difficult.

    The use of intelligence tests continued to grow throughout American schools

    over the next eighty years with the most common uses being for the identification

    of children with special needs and children with special gifts and talents (Piirto,

    2007). The use of IQ tests to investigate differences in intelligence levels of

    different ethnic groups became widely provocative with the publishing of

    Hernstein and Murray’s (1994) book entitled The Bell Curve: Intelligence and

    Class Structure in American Life. The researchers stated in their book that they

    had proven that people from minority ethnic backgrounds had lower IQs than

    white Americans. Researchers in education, such as Kincheloe, Steinberg and

    Gresson (1996), were quick to refute the allegations in their book Measured Lies:

    The Bell Curve Examined. They wrote “The Bell Curve … emerges from a

    crumbling paradigm often deemed inadequate for the study of human

    intelligence” (Kincheloe et al., 1996, p. 28). While this latter book sold widely, it

    did not impact on the growth of intelligence testing in American schools. In fact,

  • 47

    according to Piirto (2007, p. 14) “the increase in the use of aptitude,

    achievement, and personality tests has been marked.”

    Psychometric testing began to emerge in Australian schools in the early 1920s

    and is now a well-established and accepted part of testing for Australian school

    children (Hughes, 2002). The establishment of the Australian Council for

    Educational Research in 1930 provided standardised resources for psychometric

    testing to be undertaken in schools. By 1936, all Year 6 students in New South

    Wales (NSW) were administered intelligence tests to determine which form of

    secondary education best suited them. However twelve years later, a United

    Nations Educational, Scientific and Cultural Organization (UNESCO) report on

    educational psychology services across 41 countries in 1948, estimated that only

    20 psychologists were employed across all Australian school systems and were

    mostly based in NSW (Korniszewski & Mallet, 1948). Therefore, while a wealth

    of data was being collected, the analysis of the data was generally limited to

    superficial interpretation by school administrators.

    The dominance of psychometric testing in Australian and international schools

    has been driven by a widespread need of modern society to quantify individuals’

    intellectual capacity. This is evident when one considers that most students in

    Australia, and particularly those attending private schools, will not leave formal

    schooling without having undertaken at least one Intelligence Quotient (IQ) test

    and a dozen more specific intelligence tests (Bjorklund, 2000). The testing is

    sometimes undertaken internally by educators using standardised instruments

    such as the Middle Years Ability Test (Australian Council for Educational

    Research, 2005) or administered privately by organizations such as The Sydney

    Child Assessment and Testing Service (SCATS) which provides a private testing

    environment for children aged between 3 and 16 years using predominantly the

    Wechsler (1991) testing instruments.

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    Despite the widespread use and acceptance of psychometric testing in Australia

    and internationally, researchers such as Naglieri and Kaufman (2001) raised

    concerns about weaknesses in traditional IQ tests when they are used as a tool

    to determine giftedness. They said these tests were theoretically old, they were

    weak in theory and they were achievement driven. In addition, Jensen (as

    reported in Jensen & Miele, 2004) raised concerns about the blatant lack of

    understanding among users of IQ tests that has lead to the common misuse of

    data generated from such testing instruments. Richhart (2002, p. 16) alerted us

    to the impact of test practice on test scores when he said that some critics

    “contend that test scores are highly influenced by one’s test-taking competence

    and familiarity”. The existence of Core Skills Test (Queensland Studies

    Authority, 2009) preparation courses in many Queensland schools, where

    students in Year 12 practise tests from previous years to ensure they are familiar

    with the test format and timing, could be considered as evidence of the widely

    held acceptance of this viewpoint.

    While IQ results correlate positively with academic success and employability

    (Brody, 1997) and have been strongly supported by researchers such as Jensen

    (1998), other researchers argued that IQ tests are limited in what they can

    measure and that it is misleading to use an IQ score as a sole indicator of a

    child’s overall intelligence. Gardner (1999b) was one of these researchers and

    suggested that IQ tests provided at best a distorted view of an individual’s

    potential, as they clearly advantaged individuals with strengths in the linguistic

    and mathematical intelligences. Individuals with strengths in other intelligence

    areas, such as the bodily-kinaesthetic intelligence, are often neglected and

    hence do not receive an education sympathetic towards their unique form of

    intelligence. Surprisingly, Gardner is not opposed to the use of intelligence tests

    for determining intelligence of individuals. He would however, prefer that testing

    instruments were constructed to measure and evaluate all of the multiple

    intelligences. In additional to these concerns, other critics of IQ tests say they

  • 49

    are culturally biased, that is, they are based on knowledge and skills of middle-

    class individuals from majority cultures rather than being inclusive of the

    traditions, values, predominant language or experiences of minority cultures

    (Bjorklund, 2000). The concerns mentioned here are relevant to new research

    when IQ testing is used as an instrument to collect data or determine participants

    for research studies. The literature would seem to suggest that, at best, IQ test

    data can be used as an indication, rather than a definitive measure, of an

    individual’s intelligence.

    For as long as the existence of general and specific factors has been mooted,

    there have been researchers who support the existence of only a single general

    factor, or only specific factors, or both. A number of researchers supported the

    existence of specific factors but challenged the existence of a general factor.

    Debate continues about the existence of a higher-order, general intelligence

    factor that oversees and orchestrates these other cognitive factors. For example,

    Jensen (1998) is still seeking to demonstrate the factor’s existence, while others

    like Ceci (1996) proclaiming the search is “fruitless”. On the other hand, other

    researchers, such as Guilford (1988) and Sternberg (2002), have repeatedly

    attempted to disprove, without success, the pivotal influence of a g factor in

    determining intelligence of individuals however, according to Piirto (2007, p. 15),

    “general intelligence is pervasive, even in tests that purport not to measure g-

    factor intelligence.”

    Brody (2003, p. 319) adds his support to the existence of a g factor when he says

    that the “g theory is required to understand the relationships obtained by

    Sternberg and his colleagues” who were proponents of information processing

    theories of intelligence. That being said, it is now generally accepted that there is

    more to intelligence than the general intelligence factor alone (Gottfredson,

    2003). The challenge for future research is to develop a theoretical framework

    and appropriate testing tools that incorporate the notion of a g factor in

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    combination with the widely accepted multiplicity in intellectual functioning that is

    reported by researchers such as Gardner (1999b).

    Despite disagreement between researchers about the accuracy of psychometric

    testing as an accurate measure of an individual’s intelligence, or whether a

    psychometric test is a reliable instrument to measure intelligence for all

    individuals, the widespread use of IQ testing remains a feature of our present

    education systems both internationally and in Australia. Gottfredson (2003) and

    Piirto (2007) supported the use of IQ testing for the purposes of indentifying

    individuals for suitable interventions to address their particular developmental

    learning needs, however, to measure an individual’s ability within a specific

    context and in a specific area of learning, more specific testing instruments are

    required (Gardner, 1999b; Sternberg, 2000).

    2.2.3 Multiple Intelligences Theory

    In contrast to how Sternberg (1999a) emphasised the connectedness of the three

    aspects of his Triarchic Theory of Intelligence, Gardner (1999b) emphasised the

    separateness of his multiple intelligences. For him, there were up to ten unique

    intelligences that represented a modular, brain-based capacity, some of which

    were linguistic intelligence, logical-mathematical intelligence, and intrapersonal

    intelligence. His was the first theory to account for the diverse range of important

    capacities of individuals by considering a diverse range of competences and

    based his theory on a diverse range of evidence. His evidence included the

    selective damage of specific cognitive abilities following brain trauma and